7-6-39 derivative of inverse tangent and inverse secant
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To do this problem we need to remember that the derivative of
tangent inverse is 1 / 1 plus EU, in this case sqrt 16 X
squared -1 ^2.
But then we have to multiply that by the derivative of sqrt
16 X squared -1 derivative of square root is 1 / 2 square
roots of whatever was on the inside.
But then we also have to multiply by the derivative of
the inside.
SO32X plus cosecant inverse of 4X is one of our formulas and
it's going to be the -1 over the absolute value of 4X sqrt 16 X
squared -1 times the derivative of the 4X which is 4.
So now if we simplify stuff up the 32 and the two are going to
reduce to a 16X on the top down here, we're going to get 1 + 16
X squared -1 and that's going to get multiplied by sqrt 16 X
squared -1 Down here, the four in the absolute value of four
can cancel.
So we get a -1 over the absolute value of X square root sixteen X
^2 -, 1.
So now when we look at this, the +1 and -1 here are going to
cancel.
So the 16 top and bottom will reduce and one of the X's will
reduce.
So we're going to get 1 / X square root sixteen X ^2 -, 1 -,
1 over the absolute value of X square root sixteen X ^2 -, 1.
It says up here X is greater than one.
If X is greater than one, we know this is a positive, so we
can actually see that this is just going to turn into zero.